Lemma 23 [1]} Suppose that \(f \in L_2(\pi )\) , \(\mathbb {E}_{\pi }(f)=0\) , and \(-1\le f(x)\le 1\)  for every \(x \in \Omega \) . Recall the definition of spectrum \(\mathcal {S}_2\)  in Subsection REF . Then, for reversible \(\mathbf {P}\)  with spectral gap \(\gamma >0\) , for every \(0<\theta <\gamma /10\) , we have
\(\sup \mathcal {S}_2\big (e^{\theta \mathbf {D}_f}\mathbf {P}e^{\theta \mathbf {D}_f} \big )\le 1+ \frac{4V_f}{\gamma }\theta ^2 \bigg (1-\frac{100}{\gamma }\bigg )^{-1}, \qquad \mbox{and},\\\sup \mathcal {S}_2\big (e^{\theta \mathbf {D}_f}\mathbf {P}e^{\theta \mathbf {D}_f} \big )\le 1+ \big (2\sigma _{\rm {a.s}}^2+0.8 V_f\big )\theta ^2 \bigg (1-\frac{100}{\gamma }\bigg )^{-1},\) 
